Happy birthday to you!
au-GU-ri — stress on second syllable. 'Auguri' literally means 'good wishes'.
Sing or say on someone's birthday. 'Auguri' is a versatile wish used for birthdays, Christmas, Easter, name days and any celebratory occasion. Learning the full birthday song in Italian is a social asset.
'Auguri' comes from 'augurio' (omen/wish) and is Italy's all-purpose celebration word. For birthdays: 'tanti auguri a te, tanti auguri a te, tanti auguri a [name], tanti auguri a te!' — sung to the same tune as Happy Birthday. For other occasions: 'auguri!' alone suffices.
Buon compleanno!
Happy birthday!
More specific than 'auguri' — literally 'good birthday'.
Cent'anni!
May you live a hundred years!
Traditional Italian birthday wish — equivalent of 'many happy returns'.
Auguri e figli maschi!
Best wishes and male children!
Traditional, slightly old-fashioned wish — reflects historical preference for male heirs.
The Italian birthday tradition includes: the 'torta di compleanno' (birthday cake with candles), blowing the candles, the 'auguri' song, and often a party ('festa'). In many Italian families, the 'onomastico' (name day — the feast day of your saint) is celebrated nearly as importantly as the birthday itself.